Inspector Simon Mart and the stolen matchsticks

Inspector Simon Mart looked at the old sign on the door of his office with his name: Inspector Mart, S. That his parents had bestowed just one initial on him, had been one of his life long irritations. He probably should have told his parents before his birth that he wanted many birth names. But perhaps he could persuade the guys who provided the signs on the doors that his initial should precede his family name, and not come after it. Well, he had his own room and that was a benefit that should last until the next reorganization.

His manager had dumped a file on his desk. He read the attached note: ‘to be solved last month’. It was the 29th of the month, and he decided that this urgency would allow him to start with a mug of coffee and a social chat with his fellow inspectors at he coffee machine. At the coffee machine he met a new and pretty police officer and he chatted for a quarter of an hour with her. After that chat he decided to pick up this file. It contained a number of reports as well as a copy of interrogations. He summoned that a couple of matchsticks encrusted with amethysts had been stolen from the London Matchbox Museum. There were three suspects, Jim, Jack and John, all well known criminals. It was known that none of them could speak two consecutive true statements. He looked at the interrogation reports:

Jim: Jack did it. John is innocent.
Jack: John did it. Jim is innocent.
John: Jim is innocent. Jack is innocent.

You can check your solutions here